Question: $ A = \left[\begin{array}{rr}2 & -1 \\ 4 & 0\end{array}\right]$ $ B = \left[\begin{array}{rrr}0 & 0 & 1 \\ 5 & 4 & 0\end{array}\right]$ What is $ A B$ ?
Answer: Because $ A$ has dimensions $(2\times2)$ and $ B$ has dimensions $(2\times3)$ , the answer matrix will have dimensions $(2\times3)$ $ A B = \left[\begin{array}{rr}{2} & {-1} \\ {4} & {0}\end{array}\right] \left[\begin{array}{rrr}{0} & \color{#DF0030}{0} & \color{#9D38BD}{1} \\ {5} & \color{#DF0030}{4} & \color{#9D38BD}{0}\end{array}\right] = \left[\begin{array}{rrr}? & ? & ? \\ ? & ? & ?\end{array}\right] $ To find the element at any row $i$ , column $j$ of the answer matrix, multiply the elements in row $i$ of the first matrix, $ A$ , with the corresponding elements in column $j$ of the second matrix, $ B$ , and add the products together. So, to find the element at row 1, column 1 of the answer matrix, multiply the first element in ${\text{row }1}$ of $ A$ with the first element in ${\text{column }1}$ of $ B$ , then multiply the second element in ${\text{row }1}$ of $ A$ with the second element in ${\text{column }1}$ of $ B$ , and so on. Add the products together. $ \left[\begin{array}{rrr}{2}\cdot{0}+{-1}\cdot{5} & ? & ? \\ ? & ? & ?\end{array}\right] $ Likewise, to find the element at row 2, column 1 of the answer matrix, multiply the elements in ${\text{row }2}$ of $ A$ with the corresponding elements in ${\text{column }1}$ of $ B$ and add the products together. $ \left[\begin{array}{rrr}{2}\cdot{0}+{-1}\cdot{5} & ? & ? \\ {4}\cdot{0}+{0}\cdot{5} & ? & ?\end{array}\right] $ Likewise, to find the element at row 1, column 2 of the answer matrix, multiply the elements in ${\text{row }1}$ of $ A$ with the corresponding elements in $\color{#DF0030}{\text{column }2}$ of $ B$ and add the products together. $ \left[\begin{array}{rrr}{2}\cdot{0}+{-1}\cdot{5} & {2}\cdot\color{#DF0030}{0}+{-1}\cdot\color{#DF0030}{4} & ? \\ {4}\cdot{0}+{0}\cdot{5} & ? & ?\end{array}\right] $ Fill out the rest: $ \left[\begin{array}{rrr}{2}\cdot{0}+{-1}\cdot{5} & {2}\cdot\color{#DF0030}{0}+{-1}\cdot\color{#DF0030}{4} & {2}\cdot\color{#9D38BD}{1}+{-1}\cdot\color{#9D38BD}{0} \\ {4}\cdot{0}+{0}\cdot{5} & {4}\cdot\color{#DF0030}{0}+{0}\cdot\color{#DF0030}{4} & {4}\cdot\color{#9D38BD}{1}+{0}\cdot\color{#9D38BD}{0}\end{array}\right] $ After simplifying, we end up with: $ \left[\begin{array}{rrr}-5 & -4 & 2 \\ 0 & 0 & 4\end{array}\right] $